Jerzy Kocik

Krawtchouk Polynomials and Krawtchouk Matrices

Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed Sylvester-Hadamard matrices via a binary shuffling function. The underlying symmetric tensor algebra is then presented.

Clifford Algebras and Euclid’s Parametrization of Pythagorean Triples

Abstract.We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R21, whose minimal version may be conceptualized as a 4-dimensional real algebra of “kwaternions.” We observe that this makes Euclid’s parametrization the earliest appearance of the concept of spinors. We present an analogue of the “magic correspondence” for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.

BEREZIN QUANTIZATION OF THE SCHRÖDINGER ALGEBRA

We examine the Schrodinger algebra in the framework of Berezin quantization. First, the Heisenberg–Weyl and sl(2) algebras are studied. Then the Berezin representation of the Schrodinger algebra is computed. In fact, the sl(2) piece of the Schrodinger algebra can be decoupled from the Heisenberg component. This is accomplished using a special realization of the sl(2) component that is built from the Heisenberg piece as the quadratic elements in the Heisenberg–Weyl enveloping algebra. The structure of the Schrodinger algebra is revealed in a lucid way by the form of the Berezin representation.

On Geometry of Phenomenological Thermodynamics

Thanks to works of Caratheodory [4] and Gibbs [5] phenomenological thermodynamics of equilibrium (PTE) has become a standard axiomatic theory. Formulated in the general way [11,3], it reveals the structure which is universal in the sense that the later statistical and quantum statistical mechanics have not replaced it but rather serve as models of the general scheme. 1

Canonical variables and analysis on so(n, 2)

The approach of Berezin to the quantization of so(n,2) via generalized coherent states is considered in detail. A family of n commuting observables is found in which the basis for an associated Fock-type representation space is expressed. An interesting feature is that computations can be done by explicit matrix calculations in a particular basis. The basic technical tool is the Leibniz function, the inner product of coherent states.

A Porism Concerning Cyclic Quadrilaterals

We present a geometric theorem on a porism about cyclic quadrilaterals, namely, the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as for the circle.

On a Diophantine Equation That Generates All Integral Apollonian Gaskets

A remarkably simple Diophantine quadratic equation is known to generate all, Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also, occurrence of Pythagorean triples in such gaskets is discussed.

Structure of Matrix Manifolds and a Particle Model

The decomposition of matrix manifolds into homogeneous spaces of certain groups is studied in some detail. The results are applied to the derivation of the internal structure of SU(2,2)×SU(m)‐ and P4×SU(m)‐invariant particle models where the first (second) factor in the direct product represents external (internal) symmetry.

Disentangling a Triangle

This observation—which seems to be absent in all the presentations of trigonometryknown to us—proves to be a convenient tool for bringing order into the garden oftrigonometric identities. As an example, see Figure 2 for ways to visualize products ofsines and of cosines; they would perhaps please Napier in his experiments with suchproducts that eventually led him to the concept of logarithm.